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\begin{document}

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\section*{jet comonad}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra}

[[!include higher algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In a context $\mathbf{H}$ of [[differential cohesion]] with $\Im$ the [[infinitesimal shape modality]], then for any object $X\in \mathbf{H}$ the [[base change]] [[comonad]]

\begin{displaymath}
Jet_X \coloneqq i^\ast i_\ast
\end{displaymath}
for base change along the $X$-component of the [[unit of a monad|unit]] of $\Im$

\begin{displaymath}
\mathbf{H}_{/X}
  \stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}
  \mathbf{H}_{/\Im(X)}
  \,,
\end{displaymath}
may be interpreted as sending any [[bundle]] over $X$ to its [[jet bundle]].

 This characterization via base change is more or less implicit in (\hyperlink{Kock10}{Kock 10, remark 7.3.1}) (to translate from the pull-push $(p_2)_\ast p_1^\ast$ shown there, as in (\hyperlink{Deligne70}{Deligne 70}) use that in a [[topos]] the [[epimorphism]] $X \to \Im X$ is [[effective epimorphism|effective]] and then use the [[Beck-Chevalley condition]] to get the push-pull shown above.)

\begin{displaymath}
\left\{
  \itexarray{
    T^\infty X
    &\stackrel{p_1}{\longrightarrow}&
    X
    \\
    \downarrow^{\mathrlap{p_2}} &(pb)& \downarrow^{\mathrlap{i_X}}
    \\
    X &\stackrel{i_X}{\longrightarrow}& \Im X
  }
  \right\}
  \;\;
  \Rightarrow
  \;\;
  ((p_2)_\ast (p_1)^\ast \simeq (i_X)^\ast (i_X)_\ast)
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

The [[Eilenberg-Moore category]] of [[coalgebras]] over the Jet comonad has the interpretation of the category of [[partial differential equations]] with [[variables]] in $X$. The [[co-Kleisli category]] of the Jet comonad has the interpretation as being the category of bundles over $X$ with [[differential operators]] between them as morphisms (\hyperlink{Marvan86}{Marvan 86}, \hyperlink{Marvan89}{Marvan 89}).

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[topos of coalgebras over a comonad]]


\item in [[Borger's absolute geometry]] a similar base change as above is interpreted as the [[arithmetic jet space]] construction.



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

In the context of [[differential geometry]] the comonad structure on the jet bundle construction, as well as the interpretation of its EM-category as that of partial differential equations, is due to

\begin{itemize}%
\item [[Michal Marvan]], \emph{A note on the category of partial differential equations}, in \emph{Differential geometry and its applications}, Proceedings of the Conference August 24-30, 1986, Brno ([[MarvanJetComonadPDE.pdf:file]])

\end{itemize}
(Proposition 1.4 in \hyperlink{Marvan86}{Marvan 86} needs an extra ``weakened transversality'' condition on the equalizer, this is fixed in (Theorem 1.3, \hyperlink{MarvanThesis}{Marvan's thesis}). The extra condition is that the equalizer must remain an equalizer after an application of the $V$ functor, which maps fibered manifolds to their vertical tangent bundles.)

\begin{itemize}%
\item [[Michal Marvan]], thesis, 1989 ([[MarvanThesis.pdf:file]], \href{http://www.slu.cz/math/cz/lide/marvan-michal/docs/mar89-ocr.pdf}{web})


\item [[Michal Marvan]], \emph{On the horizontal cohomology with general coefficients}, 1989 (\href{http://old.math.slu.cz/People/MichalMarvan/Annotations/horizontal.php}{web announcement}, \href{http://dml.cz/dmlcz/701469}{web archive})


\item [[Michal Marvan]], section 1.1 of \emph{On Zero-Curvature Representations of Partial Differential Equations}, (1993) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.45.5631}{web})



\end{itemize}
Discussion in [[synthetic differential geometry]] is in

\begin{itemize}%
\item [[Pierre Deligne]], \emph{Equations Diff\'e{}rentielles \`a{} Points Singuliers R\'e{}guliers}, 1970


\item [[Anders Kock]], remark 7.3.1 \emph{Synthetic geometry of manifolds}, Cambridge Tracts in Mathematics 180 (2010). (\href{http://home.imf.au.dk/kock/SGM-final.pdf}{pdf})



\end{itemize}
In the context of [[algebraic geometry]] and [[D-geometry]] the comonad structure is observed in

\begin{itemize}%
\item [[Jacob Lurie]], above prop. 0.9 in \emph{Notes on crystals and algebraic D-modules} (\href{http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19%28Crystals%29.pdf}{pdf}), 2009

\end{itemize}
Discussion in [[differential cohesion]] is in

\begin{itemize}%
\item [[Igor Khavkine]], [[Urs Schreiber]], \emph{[[schreiber:Synthetic variational calculus|Synthetic geometry of differential equations: I. Jets and comonad structure]]} (\href{https://arxiv.org/abs/1701.06238}{arXiv:1701.06238})

\end{itemize}
Discussion in differentially cohesive [[homotopy type theory]] is in

\begin{itemize}%
\item [[Felix Wellen]], \emph{[[schreiber:thesis Wellen|Formalizing Cartan Geometry in Modal Homotopy Type Theory]]}, 2017

\end{itemize}
For more references see at \emph{[[jet bundle]]}.

[[!redirects jet comonads]]

[[!redirects Jet comonad]] [[!redirects Jet comonads]]



\end{document}